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Appendix  VI.

1199

In pairs as shewn, but can of course be constructed with single
cylinders. The half-speed shaft K and the valves H j are of the
usual type.

P. 898. Balancing.—Professor Dalby has devised an
ingenious graphic method for finding where additional masses,
reciprocating or rotating, should be placed to complete the
balance of an imperfectly balanced engine.

Revolving Weights.—Referring to Fig. 10570, the crank shaft
a b has a centrifugal force F, due to rotation of the crank d and
the weight w. Place a drawing-board or reference plane
transversely at any point a in the shaft. Then, transferring the
effect of F to the plane, we have

(1)  A Force F (shewn dotted) acting at a

(2)  A Moment F^r round point a.

Calculating in this manner for any number of cranks, two sets
of radial values are obtained, as at G and H, one of forces and the
other of moments. For purposes of comparison, w may be
substituted for its centrifugal force, and two polygons are drawn,
as at j and K ;

(3)  A Polygon of forces f^f^ &c.

(4)  A Polygon of moments fa^flal &c.

When both polygons  close,   there  is  perfect   balance  of  the
revolving weights.    In the figures the closures are incomplete.

Reciprocating Weights.—If these be balanced by other recipro-
cating weights, the previous construction may be exactly followed :
the weights being considered as collected at the crank pins; but
it is quite impossible to balance reciprocating by revolving weights.
Transferring their effect to a, we have, for the reciprocating
weights^

(5)  A Polygon of forces.

(6)  A Polygon of moments.

And the polygons must both close for perfect balance.

To rectify imperfect Balance.—(7) Supply a force at a equal in
magnitude and direction to .the closing line in the force polygon ;

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